Equation 13.4 on page 325 in section 13.5 is not quite correct.
Every introductory textbook I've ever seen gets this wrong, but it's wrong nevertheless, and rather obviously so. The RHS is manifestly frame-independent, while the LHS is completely frame-dependent, in accordance with the equivalence principle and in accordance with everything that was said on page 321 and half of what was said on page 320.
This is easy enough to fix: I suggest
The RHS should be called the _barogenic_ gravity, i.e. the gravity due to mass.
The LHS should be called the _framative_ gravity, i.e. the frame-relative gravity.
The framative gravity can be written with a subscript to specify what frame is intended. Equation 13.4 is nowhere near true as a general proposition, but it is a decent approximation if you write the LHS as g_@ECNR, referring to an Earth-Centered Non-Rotating reference frame. (Even then, it cannot possibly be exact, because the earth is not spherical and not homogeneous.)
In the usual terrestrial lab frame g@lab differs noticeably in direction and magnitude from g@ECNR at the same location, due to centrifugal contributions. In the frame comoving with the space station, g@station is of course wildly different from g@ECNR at the same location.
See item #107 for a catalog of related issues.
Equation 13.4 on page 325 in section 13.5 is not quite correct.
Every introductory textbook I've ever seen gets this wrong, but it's wrong nevertheless, and rather obviously so. The RHS is manifestly frame-independent, while the LHS is completely frame-dependent, in accordance with the equivalence principle and in accordance with everything that was said on page 321 and half of what was said on page 320.
This is easy enough to fix: I suggest
The framative gravity can be written with a subscript to specify what frame is intended. Equation 13.4 is nowhere near true as a general proposition, but it is a decent approximation if you write the LHS as g_@ECNR, referring to an Earth-Centered Non-Rotating reference frame. (Even then, it cannot possibly be exact, because the earth is not spherical and not homogeneous.)
In the usual terrestrial lab frame g@lab differs noticeably in direction and magnitude from g@ECNR at the same location, due to centrifugal contributions. In the frame comoving with the space station, g@station is of course wildly different from g@ECNR at the same location.